Lushness, numerical index 1 and the Daugavet property in rearrangement invariant spaces

TL;DR

This paper establishes the equivalence of lushness, the Daugavet property, and numerical index 1 in certain Banach spaces, identifying specific spaces like c0, l1, and L1[0,1] as unique examples with these properties.

Contribution

It characterizes the spaces with these properties within rearrangement invariant spaces, showing their equivalence and identifying the unique spaces that possess them.

Findings

c0, l1, and l_infinity are the only rearrangement invariant sequence spaces with these properties.

L1[0,1] is the only separable rearrangement invariant function space on [0,1] with the Daugavet property.

Lushness, the Daugavet property, and numerical index 1 are equivalent in spaces with 1-unconditional bases.

Abstract

We show that for spaces with 1-unconditional bases lushness, the alternative Daugavet property and numerical index~1 are equivalent. In the class of rearrangement invariant (r.i.)\ sequence spaces the only examples of spaces with these properties are $c_0$, $\ell_1$ and $\ell_\infty$. The only lush r.i.\ separable function space on $[0,1]$ is $L_1[0,1]$; the same space is the only r.i.\ separable function space on $[0,1]$ with the Daugavet property over the reals.